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Abstract: Systematic experiments on stress-induced phase transitions in slender polycrystalline SMA specimens have revealed two interesting instability phenomena: the coalescence of two phase transformation fronts (a narrow zone between the pure austenite and pure martensite regions, which is also called the domain wall or macroscopic interface) during the loading process leads to a sudden stress drop and that during the unloading process leads to a sudden stress jump. In order to get an insight into these two phenomena, we carry out an analytical study on stress-induced phase transitions in a thin SMA layer. We derive a quasi-2D model with a non-convex effective strain energy function while taking into account the rate-independent dissipation effect. By using a coupled series-asymptotic expansion method, we derive a one-dimensional effective expression for the total energy dissipation, which only involves the leading order term of the axial strain. The equilibrium equations are obtained by maximizing the total energy dissipation, which can be solved analytically under suitable boundary conditions. The instability is then modeled as the switch of the layer’s configuration from a nontrivial solution mode to the trivial solution mode. The limit-point instability criterion is adopted to determine the bifurcation point. Descriptions for the whole coalescence processes are provided, which capture the morphology varies of the specimen and the accompanying stress drop or jump. It is revealed that the zero limit of the thickness-length ratio can lead to the smooth switch of nontrivial modes to trivial modes with no stress drop or jump. [ABSTRACT FROM AUTHOR]

Abstract: Two-component systems (TCSs) are widely employed by bacteria to sense specific external signals and conduct an appropriate response via a phosphorylation cascade within the cell. The TCS of the agr operon in the bacterium Staphylococcus aureus forms part of a regulatory process termed quorum sensing, a cell-to-cell communication mechanism used to assess population density. Since S. aureus manipulates this “knowledge” in order to co-ordinate production of its armoury of exotoxin virulence factors required to promote infection, it is important to understand fully how this process works. We present three models of the agr operon, each incorporating a different phosphorylation cascade for the TCS since the precise nature of the cascade is not fully understood. Using numerical and asymptotic techniques we examine the effects of inhibitor therapy, a novel approach to controlling bacterial infection through the attenuation of virulence, on each of these three cascades. We present results which, if evaluated against appropriate experimental data, provide insights into the potential effectiveness of such therapy. Moreover, the TCS models presented here are of broad relevance given that TCSs are widely conserved throughout the bacterial kingdom. [Copyright &y& Elsevier]

Abstract: We consider an infinite square-cell lattice of elastic beams with a semi-infinite crack. Symmetric and antisymmetric bending modes of fracture under remote loads are examined. The related long-wave asymptotes corresponding to a continuous anisotropic bending plate are also considered. In the latter model, the symmetric mode is characterized by the square-root type singularity, whereas the antisymmetric mode results in a hyper-singular field. A solution for the continuous plate with a finite crack is also presented. These closed-form continuous solutions describe the fields in the whole plane. The main goal is to establish analytical connections between the ‘macrolevel’ state, defined by the continuous asymptote of the lattice solution, and the maximal bending moment in the crack-front beam, that is, to determine the resistance of the lattice with an initial crack to the crack advance. The solutions are obtained in the same way as for mass–spring lattices. Considering the static problems we use the discrete Fourier transform and the Wiener–Hopf technique. Monotonically distributed bending moments ahead of the crack are determined for the symmetric mode, and a self-equilibrated transverse force distribution is found for the antisymmetric mode. It is shown that in the latter case only the crack-front beam resists to the fracture development, whereas the forces in the other beams facilitate the fracture. In this way, the macrolevel fracture energy is determined in terms of the material strength. The macrolevel energy release is found to be much greater than the critical strain energy of the beam, especially in the hyper-singular mode. In both problems, it is found that among the beams surrounding the crack the crack-front beam is maximally stressed, and hence its strength defines the strength of the structure. [Copyright &y& Elsevier]

Abstract: This study deals with a linear elastic body consisting of two solids connected by a thin adhesive interphase with a small thickness . The three parts have similar elastic moduli. It is proposed to model the limit behavior of the interphase when . It has been established , using matched asymptotic expansions, that at order zero, the interphase reduces to a perfect interface, while at order one, the interphase behaves like an imperfect interface, with a transmission condition involving the displacement and the traction vectors at order zero. The perfect interface model is exactly recovered using a -convergence argument. At a higher order, a new model of imperfect interface is obtained by studying the properties of a suitable (weakly converging) sequence of equilibrium solutions. Some analytical examples are given to illustrate the results obtained. [Copyright &y& Elsevier]

Abstract: A nonuniform crack growth problem is considered for a homogeneous isotropic elastic medium subjected to the action of remote oscillatory and static loads. In the case of a plane problem, the former results in Rayleigh waves propagating toward the crack tip. For the antiplane problem the shear waves play a similar role. Under the considered conditions the crack cannot move uniformly, and if the static prestress is not sufficiently high, the crack moves interruptedly. For fracture modes I and II the established, crack speed periodic regimes are examined. For mode III a complete transient solution is derived with the periodic regime as an asymptote. Examples of the crack motion are presented. The crack speed time-period and the time-averaged crack speeds are found. The ratio of the fracture energy to the energy carried by the Rayleigh wave is derived. An issue concerning two equivalent forms of the general solution is discussed. [Copyright &y& Elsevier]

Abstract: The aim of this study is to investigate transonic flow over the axisymmetric rigid body of revolutions using matched asymptotic expansions of high Reynolds number flow. For this purpose the triple-deck model is employed. It allows to study the flow separation near a junction line where a circular cylinder is connected to a divergent conical body. It is found that in the axisymmetric transonic flow the interaction region is governed by the viscous–inviscid interaction process, where the axisymmetric Karman–Guderley equation in the inviscid part of the flow should be coupled with Prandtl’s boundary layer equations for the viscous sublayer. The coupled governing equations of the interaction region is solved using a semi-direct numerical method considering proper boundary conditions. Numerical results imply that incipience of separation may appear over the axisymmetric rigid body subject to body shape and transonic axisymmetric nature makes the flow much less prone to separation as compared to the two-dimensional flow. [Copyright &y& Elsevier]

Abstract: In this paper, we consider an initial–boundary value problem for the Korteweg–de Vries equation on the negative quarter-plane. The normalized Korteweg–de Vries equation considered is given by where x and represent dimensionless distance and time, respectively. In particular, we consider the case when the initial and boundary conditions are given by for and for . Here the initial value and we restrict attention to boundary values and in the ranges and , respectively. The method of matched asymptotic coordinate expansions is used to obtain the large- asymptotic structure of the solution to this problem, which exhibits the formation of a dispersive shock wave when . We also present detailed numerical simulations of the full initial–boundary value problem which support the asymptotic analysis presented. A brief discussion is also given of the large- asymptotic structure to this problem when and . [Copyright &y& Elsevier]

Abstract: A new mathematical model of cell signalling for a two-ligand G-protein coupled receptor (GPCR) system is presented. This model extends the single-ligand cubic ternary complex to account for the possibility of an agonist and an antagonist competing for receptor sites. The G-protein cycle is included, and signalling as far as the dissociated subunit is considered. Numerical simulations are performed, and the effects on the system dynamics, such as peak and plateau behaviour, of antagonist “stickiness”, and of the doses of agonist and antagonist, are discussed. Under certain parameter regimes, the plateau response is subject to surmountable antagonism, while the peak response is subject to insurmountable antagonism. The numerical results reveal responses evolving over a number of time-scales. An asymptotic analysis is presented which identifies dominant reactions and gives leading order solutions over these various time-scales, for a number of parameter regimes. [Copyright &y& Elsevier]

Abstract: Material failure by crack propagation essentially involves a concentration of large displacement-gradients near a crack''s tip, even at scales where no irreversible deformation and energy dissipation occurs. This physical situation provides the motivation for a systematic gradient expansion of general nonlinear elastic constitutive laws that goes beyond the first order displacement-gradient expansion that is the basis for linear elastic fracture mechanics (LEFM). A weakly nonlinear fracture mechanics theory was recently developed by considering displacement-gradients up to second order. The theory predicts that, at scales within a dynamic lengthscale from a crack''s tip, significant displacements and displacement-gradient contributions arise. Whereas in LEFM the singularity generates an unbalanced force and must be discarded, we show that this singularity not only exists but is also necessary in the weakly nonlinear theory. The theory generates no spurious forces and is consistent with the notion of the autonomy of the near-tip nonlinear region. The J-integral in the weakly nonlinear theory is also shown to be path-independent, taking the same value as the linear elastic J-integral. Thus, the weakly nonlinear theory retains the key tenets of fracture mechanics, while providing excellent quantitative agreement with measurements near the tip of single propagating cracks. As is consistent with lengthscales that appear in crack tip instabilities, we suggest that this theory may serve as a promising starting point for resolving open questions in fracture dynamics. [Copyright &y& Elsevier]

Abstract: In this paper we address the vector problem of a 2D half-plane interfacial crack loaded by a general asymmetric distribution of forces acting on its faces. It is shown that the general integral formula for the evaluation of stress intensity factors, as well as high-order terms, requires both symmetric and skew-symmetric weight function matrices. The symmetric weight function matrix is obtained via the solution of a Wiener–Hopf functional equation, whereas the derivation of the skew-symmetric weight function matrix requires the construction of the corresponding full field singular solution. The weight function matrices are then used in the perturbation analysis of a crack advancing quasi-statically along the interface between two dissimilar media. A general and rigorous asymptotic procedure is developed to compute the perturbations of stress intensity factors as well as high-order terms. [Copyright &y& Elsevier]

Abstract: We derive solutions of the Kirchhoff equations for a knot tied on an infinitely long elastic rod subjected to combined tension and twist, and held at both endpoints at infinity. We consider the case of simple (trefoil) and double (cinquefoil) knots; other knot topologies can be investigated similarly. The rod model is based on Hookean elasticity but is geometrically nonlinear. The problem is formulated as a nonlinear self-contact problem with unknown contact regions. It is solved by means of matched asymptotic expansions in the limit of a loose knot. We obtain a family of equilibrium solutions depending on a single loading parameter (proportional to applied twisting moment divided by square root of pulling force), which are asymptotically valid in the limit of a loose knot, . Without any a priori assumption, we derive the topology of the contact set, which consists of an interval of contact flanked by two isolated points of contacts. We study the influence of the applied twist on the equilibrium. [Copyright &y& Elsevier]

Abstract: In this paper, the most popular proposed mechanism for activation of G-protein coupled receptors (GPCRs) – the shuttling mechanism – is modelled mathematically. An asymptotic analysis of this model clarifies the dynamics of the system in the presence of a drug, in particular identifying which reactions dominate during the different timescales. The modelling also reveals challenging behaviour in the form of a peak response. This new mechanism gives simple explanations for complex, possibly misunderstood, behaviour. [Copyright &y& Elsevier]

Abstract: The paper concerns intrinsic low-dimensional manifold (ILDM) method suggested in [Maas U, Pope SB. Simplifying chemical kinetics: intrinsic low-dimensional manifolds in composition space, combustion and flame 1992;88:239–64] for dimension reduction of models describing kinetic processes. It has been shown in a number of publications [Goldfarb I, Gol’dshtein V, Maas U. Comparative analysis of two asymptotic approaches based on integral manifolds. IMA J Appl Math 2004;69:353–74; Kaper HG, Kaper TJ, Asymptotic analysis of two reduction methods for systems of chemical reactions. Phys D 2002;165(1–2):66–93; Rhodes C, Morari M, Wiggins S. Identification of the low order manifolds: validating the algorithm of Maas and Pope. Chaos 1999;9(1):108–23] that the ILDM-method works successfully and the intrinsic low-dimensional manifolds belong to a small vicinity of invariant slow manifolds. The ILDM-method has a number of disadvantages. One of them is appearance of so-called “ghost”-manifolds, which do not have connection to the system dynamics [Borok S, Goldfarb I, Gol’dshtein V. “Ghost” ILDM – manifolds and their discrimination. In: Twentieth Annual Symposium of the Israel Section of the Combustion Institute, Beer-Sheva, Israel; 2004. p. 55–7; Borok S, Goldfarb I, Gol’dshtein V. About non-coincidence of invariant manifolds and intrinsic low-dimensional manifolds (ILDM). CNSNS 2008;71:1029–38; Borok S, Goldfarb I, Gol’dshtein V, Maas U. In: Gorban AN, Kazantzis N, Kevrekidis YG, Ottinger HC, Theodoropoulos C, editors. “Ghost” ILDM-manifolds and their identification: model reduction and coarse-graining approaches for multiscale phenomena. Berlin–Heidelberg–New York: Springer; 2006. p. 55–80; Borok S, Goldfarb I, Gol’dshtein V. On a modified version of ILDM method and its asymptotic analysis. IJPAM 2008; 44(1): 125–50; Bykov V, Goldfarb I, Gol’dshtein V, Maas U. On a modified version of ILDM approach: asymptotic analysis based on integral manifolds. IMA J Appl Math 2006;71:359–82; Flockerzi D. Tutorial: intrinsic low-dimensional manifolds and slow attractors. Magdeburg: Max-Planck-Institut; 2001–2005. ; Flockerzi D, Heineken W. Comment on “Identification of low order manifolds: validating the algorithm of Maas and Pope”. Chaos 1999;9:108–23; Flockerzi D, Heineken W. Comment on “Identification of low order manifolds: validating the algorithm of Maas and Pope”. Chaos 2006;16:048101]. The present work studies the causes for the “ghost” manifolds appearance for the case of a two-dimensional singularly perturbed system. [Copyright &y& Elsevier]

Abstract: We derive a continuum model for the Peach–Koehler force on dislocations in a slip plane. To represent the dislocations, we use the disregistry across the slip plane, whose gradient gives the density and direction of the dislocations. The continuum model is derived rigorously from the Peach–Koehler force on dislocations in a region that contains many dislocations. The resulting continuum model can be written as the variation of an elastic energy that consists of the contribution from the long-range elastic interaction of dislocations and a correction due to the line tension effect. [Copyright &y& Elsevier]

Abstract: We study the interaction between the buildings of a city through a fully coupling structure–soil–structure model. The main issues are to emphasize the collective behavior of the buildings during a seismic excitation (city effect) and to compute the global response of the city. The anti-plane shearing of an elastic half space with a part of its boundary formed by the rigid foundations of the buildings is considered. The buildings are modeled as elastic springs which relate two concentrated masses. The associated mathematical problem is a partial differential equation coupled with an ordinary differential equation through a special class of boundary conditions. To focus on the coupling structure–soil–structure (city frequencies) the associated spectral problem is rewritten in terms of a nonlinear scalar equation involving the eigenvalues of a symmetric matrix. A numerical approach, involving an integral method, is proposed to numerically investigate the city effect. The asymptotic limit of the principal eigenvalue with respect to the ratio between the width of the building and the width of the city points out the principal frequency of the city. This value does not depended on the number of buildings which compose the city but it is specific to each city via the specific properties of the buildings. The collective behavior of the buildings during a seismic excitation is emphasized through the shape of the first eigenfunction. Finally a numerical illustration of a the propagation in the city of the impact on the top of one of the buildings (a 9/11 type event) is presented. [Copyright &y& Elsevier]

• We give a (wider) vision of the state of the art with new references commented in the introduction. • We obtain a reduced model for elliptic membrane shells in normal damped response contact in thermoelasticity. • We show existence and uniqueness of both the three-dimensional and the two-dimensional limit problems. • We use the asymptotic expansion method to obtain the model. • We justify the validity of the model by providing a strong convergence result. The purpose of this paper is twofold. We first provide the mathematical analysis of a dynamic contact problem in thermoelasticity, when the contact is governed by a normal damped response function and the constitutive thermoelastic law is given by the Duhamel-Neumann relation. Under suitable hypotheses on data and using a Faedo-Galerkin strategy, we show the existence and uniqueness of solution for this problem. Then, we study the particular case when the deformable body is, in fact, a shell and use asymptotic analysis to study the convergence to a 2D limit problem when the thickness tends to zero. [ABSTRACT FROM AUTHOR]

• A reverse-time nonlocal nonlinear self-dual network equation is proposed. • Discrete nonlocal version of the N-fold Darboux transformation is constructed. • Unstable and stable soliton solutions are derived and shown graphically. • It is shown that individual components show instability whereas the potential terms present stable structures. Inspired by the idea of Ablowitz and Musslimani, a reverse-time nonlocal nonlinear self-dual network equation is proposed under certain nonlocal symmetry reduction. N -fold Darboux transformation technique is used to construct discrete unstable and stable multi-soliton solutions in zero seed background for this system. Based on the asymptotic and graphic analysis, structures of one-, two- and three-soliton solutions are shown graphically. It is found that the individual components in this nonlocal equation show unstable temporal soliton structures whereas the combined potential terms exhibit stable soliton structures. Dynamical behaviors for one-soliton solutions are investigated via numerical simulations. It is worth noting that the solutions of this nonlocal equation are clearly different from those of its corresponding local part. Results obtained in this paper may be helpful for the explanation of the propagation of electrical signals. [ABSTRACT FROM AUTHOR]

Abstract: The topological derivative gives the sensitivity of the problem when the domain under consideration is perturbed by the introduction of a hole. Alternatively, this same concept can also be used to calculate the sensitivity of the problem when, instead of a hole, a small inclusion is introduced at a point in the domain. In the present paper we apply the Topological-Shape Sensitivity Method to obtain the topological derivative of inclusion in two-dimensional linear elasticity, adopting the total potential energy as the cost function and the equilibrium equation as a constraint. For the sake of completeness, initially we present a brief description of the Topological-Shape Sensitivity Method. Then, we calculate the topological derivative for the problem under consideration in two steps: firstly we perform the shape derivative and next we calculate the limit when the perturbation vanishes using classical asymptotic analysis around a circular inclusion. In addition, we use this information as a descent direction in a topology design algorithm which allows to simultaneously remove and insert material. Finally, we explore this feature showing some numerical experiments of structural topology design within the context of two-dimensional linear elasticity problem. [Copyright &y& Elsevier]

This paper provides experimental confirmation of the opening asymptotes that have been predicted to develop at the tip of fluid-driven cracks propagating in impermeable brittle elastic media. During propagation of such cracks, energy is dissipated not only by breaking of material bonds ahead of the tip but also by flow of viscous fluid. Theoretical analysis based on linear elastic fracture mechanics and lubrication theory predicts a complex multiscale asymptotic behavior of the opening in the tip region, which simplifies either as or as power law of the distance from the tip depending on whether the dominant mechanism of energy dissipation is bond breaking or viscous flow. The laboratory experiments entail the propagation of penny-shaped cracks by injection of glycerin or glucose based solutions in polymethyl methacrylate (PMMA) and glass specimens subjected to confining stresses. The full-field opening is measured from analysis of the loss of intensity as light passes through the dye-laden fluid that fills the crack. The experimental near-tip opening gives excellent agreement with theory and therefore confirms the predicted multi-scale tip asymptotics. [Copyright &y& Elsevier]

Abstract: Current knowledge of microbial mutation rates was accumulated largely by means of fluctuation experiments. A mathematical model describing the cell dynamics in a fluctuation experiment is indispensable to the estimation of mutation rates through fluctuation experiments. In almost six decades the model formulated by Lea and Coulson dominated in research and application, although the model formulated by Bartlett is generally believed to describe the cell dynamics more faithfully. The neglect of the Bartlett formulation was mainly due to mathematical difficulties. The present investigation overcomes some of these difficulties, thereby paving the way for the application of the Bartlett formulation in estimating mutation rates. Specifically, the article offers an algorithm for computing the distribution function of the number of mutants under the Bartlett formulation. The article also provides algorithms for computing point and interval estimates of mutation rates that are based on the maximum-likelihood principle. In addition, the article examines and compares the asymptotic behavior of the distributions of the number of mutants under the two formulations. [Copyright &y& Elsevier]

Abstract: The effective elastic properties of periodic fibre-reinforced media with complex microstructure are determined by the method of asymptotic homogenization via a novel solution to the cell problem. The solution scheme is ideally suited to materials with many fibres in the periodic cell. In this first part of the paper we discuss the theory for the most general situation—N arbitrarily anisotropic fibres within the periodic cell. For ease of exposition we then restrict attention to isotropic phases which results in a monoclinic composite material with 13 effective moduli and expressions for each of these are determined. In the second part of this paper we shall discuss results for a variety of specific microstructures. [Copyright &y& Elsevier]

Abstract: Anti-plane and plane-strain, time-harmonic, small-amplitude vibrations of an elastic layer on an elastic half space are considered, superimposed upon a state of finite, uniform stress and strain. A (compressible) elastic material is considered, orthotropic with orthotropy axes aligned parallel and orthogonal both to the layer and the prestress principal directions. A non-uniform mass density is assumed in the layer. A formal long-wave asymptotic solution is derived under the assumptions of high contrast between the stiffnesses of the layer and the half space and between certain prestress components and the current elastic shear modulus. It is shown that (i) the layer asymptotically behaves as a beam subject to transversal and axial vibrations; (ii) the response of the half space can be found in a closed-form, under the assumption of plane wave motion (which becomes consistent when the density of the layer is uniform), otherwise it is represented by a hypersingular integral equation; (iii) if the nonlocality introduced by the hypersingular integral equation is restricted to an influence area of finite extent, the integral can be analytically approximated, so that a Winkler-type spring model representing the half space is rigorously derived. For uniform density of the layer, the constants defining the spring model are given as functions of the prestress and anisotropy parameters of the half space; and, finally, (iv) the asymptotic solution provides new analytical expressions for incremental displacement of the layer, which, compared to the exact numerical solution (also included), are shown to perform quite well, even for values of parameters much beyond the limits imposed by the asymptotic analysis. The asymptotic analysis allows us to explore, for the first time, dynamic properties of a periodic layer bonded to an elastic half space and subject to a uniform prestress state. We find that the system exhibits band gaps (ranges of forbidden frequencies) and that the prestress can be used as a parameter tuning the filtering properties of the structure, an effect which may have important consequences in the design of resonant devices. [Copyright &y& Elsevier]

Abstract: We study the buckling of a compressed thin elastic film bonded to a compliant substrate. An asymptotic solution of the equations for a plate on an elastic foundation is obtained in the limit of large residual stress in the film. In this limit, the film''s shape is given by a popular origami folding, the Miura-ori, and is composed of parallelograms connected by dihedral folds. This asymptotic solution corresponds to the herringbone patterns reported previously in experiments: the crests and valleys of the pattern define a set of parallel, sawtooth-like curves. The kink angle obtained when observing these crests and valleys from above are shown to be right angles under equi-biaxial loading, in agreement with the experiments. The absolute minimum of energy corresponds to a pattern with very slender parallelograms; in the experiments, the wavelength is instead selected by the history of applied load. [Copyright &y& Elsevier]

Abstract: We study the buckling of a thin compressed elastic film bonded to a compliant substrate. We focus on a family of buckling patterns, such that the film profile is generated by two functions of a single variable. This family includes the unbuckled configuration, the classical primary mode made of straight stripes, as well the pattern with undulating stripes obtained by a secondary instability investigated in the first companion paper, and the herringbone pattern studied in last companion paper. A simplified buckling model relevant for the analysis of these patterns is introduced. It is solved analytically for moderate or for large residual compressive stress in the film. Numerical simulations are presented, based on an efficient implementation. Overall, the analysis provides a global picture for the formation of herringbone patterns under increasing residual stress. The film shape is shown to converge at large load to a developable shape with ridges. The wavelength of the pattern, selected in a first place by the primary buckling bifurcation, is frozen during the subsequent increase of loading. [Copyright &y& Elsevier]

Abstract: The buckling of a thin elastic film bound to a compliant substrate is studied: we analyze the different patterns that arise as a function of the biaxial residual compressive stress in the film. We first clarify the boundary conditions to be used at the interface between film and substrate. We carry out the linear stability analysis of the classical pattern made of straight stripes, and point out secondary instabilities leading to the formation of undulating stripes, varicose, checkerboard or hexagonal patterns. Straight stripes are found to be stable in a narrow window of load parameters only. We present a weakly nonlinear post-buckling analysis of these patterns: for equi-biaxial residual compression, straight wrinkles are never stable and square checkerboard patterns are found to be optimal just above threshold; for anisotropic residual compression, straight wrinkles are present above a primary threshold and soon become unstable with respect to undulating stripes. These results account for many of the previously published experimental or numerical results on this geometry. [Copyright &y& Elsevier]

Abstract: The self-force and effective mass of a moving dislocation in a generally accelerating motion are explicitly obtained on the basis of a surface-independent dynamic J-integral. Logarithmic singularities due to non-zero acceleration result in divergent integrals in the dynamic J-integral, which are treated by smearing out the dislocation core (ramp-core) and by regularizing in the sense of distributions, both coinciding in the leading terms. [Copyright &y& Elsevier]

Abstract: One studies the evolution in time of the deformation of the front of a semi-infinite 3D interface crack propagating quasistatically in an infinite heterogeneous elastic body. The fracture properties are assumed to be lower on the interface than in the materials so that crack propagation is channelled along the interface, and to vary randomly within the crack plane. The work is based on earlier formulae which provide the first-order change of the stress intensity factors along the front of a semi-infinite interface crack arising from some small but otherwise arbitrary in-plane perturbation of this front. The main object of study is the long-time behavior of various statistical measures of the deformation of the crack front. Special attention is paid to the influences of the mismatch of elastic properties, the type of propagation law (fatigue or brittle fracture) and the stable or unstable character of 2D crack propagation (depending on the loading) upon the development of this deformation. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to determine , the overall homogenized Love–Kirchhoff strength domain of a rigid perfectly plastic multi-layered plate, and to study the relationship between the 3D and the homogenized Love–Kirchhoff plate limit analysis problems. In the Love–Kirchhoff model, the generalized stresses are the in-plane (membrane) and the out-of-plane (flexural) stress field resultants. The homogenization method proposed by Bourgeois [1997. Modélisation numérique des panneaux structuraux légers. Ph.D. Thesis, University Aix-Marseille] and Sab [2003. Yield design of thin periodic plates by a homogenization technique and an application to masonry wall. C. R. Méc. 331, 641–646] for in-plane periodic rigid perfectly plastic plates is justified using the asymptotic expansion method. For laminated plates, an explicit parametric representation of the yield surface is given thanks to the -function (the plastic dissipation power density function) that describes the local strength domain at each point of the plate. This representation also provides a localization method for the determination of the 3D stress components corresponding to every generalized stress belonging to . For a laminated plate described with a yield function of the form , where is a positive even function of the out-of-plane coordinate and is a convex function of the local stress , two effective constants and a normalization procedure are introduced. A symmetric sandwich plate consisting of two Von-Mises materials ( in the skins and in the core) is studied. It is found that, for small enough contrast ratios (), the normalized strength domain is close to the one corresponding to a homogeneous Von-Mises plate [Ilyushin, A.-A., 1956. Plasticité. Eyrolles, Paris]. [Copyright &y& Elsevier]

Abstract: A weight function matrix is developed for obtaining the stress singularity coefficients at the edge of a plane crack, moving uniformly at an intersonic speed while subjected to arbitrary shear loading. This is then utilised for deriving, to first order, the perturbations of these coefficients associated with a small spatially and temporally varying perturbation of its edge. The perturbation solution is employed, in conjunction with a simple fracture criterion, to investigate the stability of a uniformly moving intersonic crack, subjected to following loads. [Copyright &y& Elsevier]

Abstract: In this paper, we study a three-dimensional axisymmetric boundary-value problem of a slender cylinder composed of a nonlinearly elastic material subjected to an axial force. Starting from the field equations, after a transformation and proper scalings, we identify a small variable and two small parameters, which characterize the present problem. Then, by an approach involving compound series-asymptotic expansions, a nonlinear ODE is derived, which governs the axial strain (the first-term in the series expansion). By imposing the zero radial displacement conditions at two ends, we manage to get the analytical solution of the axial strain, from which all other physical quantities can be deduced and thus the three-dimensional displacement field can be determined. Graphical results are presented, which show that there are two boundary layers near the two ends while the middle part is in a state of almost uniform extension. The asymptotic structure of the analytical solution is derived, which offers clear explanations to the structure of the deformed configuration and shows that the thickness of both boundary layers is of the order of the radius. We also point out the relevance of the present results to the St. Venant’s problem. In particular, we obtain the explicit uniformly-valid exponentially small error term, when the obtained deformed configuration is compared to the configuration of a uniform extension. [Copyright &y& Elsevier]

Abstract: The paper addresses the problem of a semi-infinite plane crack along the interface between two isotropic half-spaces. Two methods of solution have been considered in the past: Lazarus and Leblond [1998a. Three-dimensional crack-face weight functions for the semi-infinite interface crack-I: variation of the stress intensity factors due to some small perturbation of the crack front. J. Mech. Phys. Solids 46, 489–511, 1998b. Three-dimensional crack-face weight functions for the semi-infinite interface crack-II: integrodifferential equations on the weight functions and resolution J. Mech. Phys. Solids 46, 513–536] applied the “special” method by Bueckner [1987. Weight functions and fundamental fields for the penny-shaped and the half-plane crack in three space. Int. J. Solids Struct. 23, 57–93] and found the expression of the variation of the stress intensity factors for a wavy crack without solving the complete elasticity problem; their solution is expressed in terms of the physical variables, and it involves five constants whose analytical representation was unknown; on the other hand, the “general” solution to the problem has been recently addressed by Bercial-Velez et al. [2005. High-order asymptotics and perturbation problems for 3D interfacial cracks. J. Mech. Phys. Solids 53, 1128–1162], using a Wiener-Hopf analysis and singular asymptotics near the crack front. The main goal of the present paper is to complete the solution to the problem by providing the connection between the two methods. This is done by constructing an integral representation for Lazarus–Leblond''s weight functions and by deriving the closed form representations of Lazarus–Leblond''s constants. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to further confirm some interpretations we made previously from materials or rock structure in term of the requirement of the introduction of complex critical exponents, where catastrophic brittle fracture is considered as a kind of second-order phase transition by analogy with percolation phenomenon. We propose here, using acoustic emission measurement data, a more complete experimental validation to support our previous conjecture that, “the higher the grain size and power supply, the longer range the interaction, and therefore the higher the imaginary part of complex critical exponents,” [Moura, A., Lei, X.L, Nishizawa, O., 2005. Prediction scheme for the catastrophic failure of highly loaded brittle materials or rocks. J. Mech. Phys. Solids 53(11), 2435–2455]. [Copyright &y& Elsevier]

Abstract: A new multivariate non-parametric regression method is considered, which is an extension of PLS1 regression using the multivariate local polynomial regression framework. A theoretical framework that can be used in order to study the asymptotic properties of the estimator is proposed, and the method is implemented on a real data set, made up of seasonal amounts of rainfall in the north of the Nordeste region of Brazil, to be explained by climatic variables. The performances of the three methods of PLS1 regression, multivariate local regression and local PLS1 regression are compared by means of cross-validation and the use of a validation period. All calculations have been implemented in the S-Plus package. The results confirm the good properties of local PLS1 regression. [Copyright &y& Elsevier]

Abstract: The equilibrium morphology of a strained island on an elastic substrate is determined. The island is assumed to partially wet the substrate (Volmer–Weber growth) and thus makes a non-zero contact angle with the surface. Both isotropic and anisotropic misfit strain are allowed. Two- and three-dimensional equilibrium island shapes are determined by using expressions for the elastic strain energy in the small-slope approximation. In this limit, the problem can be reduced to a singular integral-differential equation for the island thickness. We find that when there is a non-zero contact angle, all island shapes, for a given ratio of the elastic stress to surface energy, attain a form that is independent of the specific contact angle under an appropriate scaling. We show that for islands with non-zero contact angles, as the island volume increases, the shape approaches the geometry of a completely wetting island. But when the volume decreases, these islands approach a point while islands with a zero contact angle, approach a finite length line segment of zero volume. Multiple-hump equilibrium shapes are found. Single-humped islands are shown to have a lower chemical potential than multiple-humped islands, implying that they are the most stable. This conclusion is shown to be consistent with a stability analysis of the two-dimensional case. The effects of a tetragonal misfit strain on the three-dimensional island shape is investigated. [Copyright &y& Elsevier]

Abstract: The WATERWAVES program suite performs complex scattering calculations by propagating a wave packet in a complex, full-dimensional potential for non-rotating () but vibrating triatomic molecules. Potential energy and decay probability surfaces must be provided. Expectation values of geometric quantities can be calculated, which are useful for following the wave packet motion. The programs use a local complex potential approximation (LCP) for the Hamiltonian and Jacobi coordinates. The bottleneck of the calculation is the application of each term of the Hamiltonian to the wave packet. To solve this problem the programs use a different representation for each term: normalized associated Legendre polynomials as a functional basis for the angular kinetic term and an evenly spaced grid for the radial kinetic term yielding a fully point-wise representation of the wave functions. The potential term is treated using an efficient Discrete Variable Representation (DVR) being diagonal in the coordinate representation. The radial kinetic term uses a fast Fourier transform (FFT) to obtain an operator which is diagonal in the momentum space. To avoid artificial reflection at the boundaries of the grid a complex absorbing potential is included for calculating continuum quantities. Asymptotic analysis is performed to obtain scattering observables such as cross sections and other dynamical properties. Program summary: Program title: WATERWAVES Catalogue identifier:ADXT_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADXT_v1_0 Program obtainable from: CPC Program Library, Queen''s University of Belfast, N. Ireland Licensing provisions: Freely available from CPC Programming language: Fortran 77 Computer(s) for which the program has been designed: PC Operating system(s) for which the program has been designed: Linux RAM required to execute with typical data: case dependent: test run requires 976 024 kB No. of bytes in distributed program, including test data, etc.:11 262 681 No. of lines in distributed program, including test data, etc.:2 510 113 Distribution format:tar.gz Number of processors used: 1 External routines/libraries used: None CPC Program Library subprograms used: None Nature of problem: The WATERWAVES program suite performs complex scattering calculations for calculating the nuclear dynamics subsequent to excitation by, for example, electron or photon impact and leading to dissociative attachment or vibrational excitation. This program treats in a fully multidimensional way the nuclear motion based on the use of complex potential surfaces of three-atom systems. Complex potential energy surfaces have to be provided from fixed-nuclei calculations. Solution method: By propagating a wave packet in a complex, full-dimensional potential for non-rotating () but vibrating triatomic molecules, expectation values of geometric quantities can be calculated, which are useful for following the wave packet motion. The programs use a local complex potential approximation (LCP) for the Hamiltonian and Jacobi coordinates. Potential energy and decay probability surfaces must be provided. To solve this problem the programs use a different representation for each term. To avoid artificial reflection at the boundaries of the grid a complex absorbing potential is included for calculating continuum quantities. Asymptotic analysis is performed to obtain scattering observables such as cross sections and other dynamical properties. Additional comments: Program asymptot.f for analysing results, test data and shell scripts to make a movie of the results. Running time: Case dependent: test run requires about 200 minutes on a 3.00 GHz Intel Pentium 4 machine provided with 2 060 044 kB of memory. [Copyright &y& Elsevier]

Abstract: Wave modes in a straight cylindrical duct with a locally reacting boundary and a steady subsonic axial flow are investigated. The duct modes are separated into ordinary duct modes and surface modes confined to a neighbourhood of the boundary. Previous asymptotic results of Rienstra for the surface modes assume that the dimensionless frequency ω is large, and that the azimuthal order m ≪ ω. In this paper, these results are generalized to arbitrary values of m, as applicable to rotor-alone noise in aeroengines where m is a multiple of the number of blades and is typically O(ω). A dimensionless number λ is found to govern the surface modes’ behaviour, and qualitatively different behaviour is seen for λ in four distinct regions, separated by critical values depending only on the steady flow Mach number. For aeroacoustically relevant parameters, our generalized asymptotics are shown to provide a distinctly better approximation to the full equations than previous approximations. [Copyright &y& Elsevier]

Abstract: This paper applies the stability theory of crystalline solids presented in the companion paper (Part I) to the study of martensitic transformations found in shape memory alloys (SMA''s). The focus here is on temperature-induced martensitic transformations of bi-atomic crystals under stress-free loading conditions. A set of temperature-dependent atomic potentials and a multilattice description are employed to derive the energy density of a prototypical SMA ( cubic austenite crystal). The bifurcation and stability behavior are then investigated with respect to two stability criteria (Cauchy–Born (CB) and phonon). Using a 4-lattice description five different equilibrium crystal structures are predicted: cubic, tetragonal, orthorhombic, Cmmm orthorhombic, and monoclinic. For our chosen model only the and equilibrium paths have stable segments which satisfy both the CB- and phonon-stability criteria. These stable segments overlap in temperature indicating the possibility of a hysteretic temperature-induced proper martensitic transformation. The and crystal structures are common in SMA''s and therefore the simulated jump in the deformation gradient at a temperature for which both crystals are stable is compared to experimental values for NiTi, AuCd, and CuAlNi. Good agreement is found for the two SMA''s which have cubic to orthorhombic transformations (AuCd and CuAlNi). [Copyright &y& Elsevier]

Abstract: Many crystalline materials exhibit solid-to-solid martensitic phase transformations in response to certain changes in temperature or applied load. These martensitic transformations result from a change in the stability of the material''s crystal structure. It is, therefore, desirable to have a detailed understanding of the possible modes through which a crystal structure may become unstable. The current work establishes the connections between three crystalline stability criteria: phonon-stability, homogenized-continuum-stability, and the presently introduced Cauchy-Born-stability criterion. Stability with respect to phonon perturbations, which probe all bounded perturbations of a uniformly deformed specimen under “hard-device” loading (i.e., all around displacement type boundary conditions) is hereby called “constrained material stability”. A more general “material stability” criterion, motivated by considering “soft” loading devices, is also introduced. This criterion considers, in addition to all bounded perturbations, all “quasi-uniform” perturbations (i.e., uniform deformations and internal atomic shifts) of a uniformly deformed specimen, and it is recommend as the relevant crystal stability criterion. [Copyright &y& Elsevier]

Abstract: Due to broadly distributed heterogeneities, the multistep character of the development of random damage in brittle materials or rocks, reflecting the discrete nature of the fracturing process, enables us to compare the diffusion of damages with the percolation phenomenon. Based on a discrete scale hierarchy, the fracturing process leading to material failure is then identified with a second-order phase transition, enabling prediction. In this connection, this paper presents a prediction scheme for the catastrophic failure or the lifetime of highly loaded materials or rocks. An experimental measurement method and a modelling of bi-phase materials or rocks are proposed. Although this scheme is introduced in the context of materials science, it also works in rock for the prediction of large natural earthquakes and some artificially induced earthquakes and rock bursts due to damming and mining. Results with strong forecasting potential explain for the first time the origin of complex critical exponents through a structural parameter. [Copyright &y& Elsevier]

Abstract: We consider chains of dimensionless masses connected by breakable bistable links. A non-monotonic piecewise linear constitutive relation for each link consists of two stable branches separated by a gap of zero resistance. Mechanically, this model can be envisioned as a ”twin-element” structure which consists of two links (rods or strands) of different lengths joined by the ends. The longer link does not resist to the loading until the shorter link breaks. We call this construction the waiting link structure. We show that the chain of such strongly non-linear elements has an increased in-the-large stability under extension in comparison with a conventional chain, and can absorb a large amount of energy. This is achieved by two reasons. One is an increase of dissipation in the form of high-frequency waves transferring the mechanical energy to heat; this is a manifestation of the inner instabilities of the bonds. The other is delocalization of the damage of the chain. The increased stability is a consequence of the distribution of a partial damage over a large volume of the body instead of its localization, as in the case of a single neck formation in a conventional chain. We optimize parameters of the structure in order to improve its resistance to a slow loading and show that it can be increased significantly by delocalizing a damage process. In particular, we show that the dissipation is a function of the gap between the stable branches and find an optimal gap corresponding to maximum energy consumption under quasi-static extension. The results of numerical simulations of the dynamic behavior of bistable chains show that these chains can withstand without breaking the force which is several times larger than the force sustained by a conventional chain. The formulation and results are also related to the modelling of compressive destruction of a porous material or a frame construction which can be described by a two-branched diagram with a large gap between the branches. We also consider an extension of the model to multi-link chain that could imitate plastic behavior of material. [Copyright &y& Elsevier]

Abstract: Solutions for the traction distributions and corresponding sub-surface state of stress adjacent to the edge of an incomplete contact suffering partial slip are found. The effects of frictional shakedown and a synchronously varying in-plane tension on the solution are found in closed form. The value of the asymptote, and its characterisation by just three independent parameters is illustrated by applying it to the finite problem of a rigid, tilted punch pressed onto a half-plane, and suffering partial slip induced by the application of in-plane tension. [Copyright &y& Elsevier]

The differential equations governing transfer and stiffness matrices and acoustic impedance for a functionally graded generally anisotropic magneto-electro-elastic medium have been obtained. It is shown that the transfer matrix satisfies a linear 1st order matrix differential equation, while the stiffness matrix satisfies a nonlinear Riccati equation. For a thin nonhomogeneous layer, approximate solutions with different levels of accuracy have been formulated in the form of a transfer matrix using a geometrical integration in the form of a Magnus expansion. This integration method preserves qualitative features of the exact solution of the differential equation, in particular energy conservation. The wave propagation solution for a thick layer or a multilayered structure of inhomogeneous layers is obtained recursively from the thin layer solutions. Since the transfer matrix solution becomes computationally unstable with increase of frequency or layer thickness, we reformulate the solution in the form of a stable stiffness-matrix solution which is obtained from the relation of the stiffness matrices to the transfer matrices. Using an efficient recursive algorithm, the stiffness matrices of the thin nonhomogeneous layer are combined to obtain the total stiffness matrix for an arbitrary functionally graded multilayered system. It is shown that the round-off error for the stiffness-matrix recursive algorithm is higher than that for the transfer matrices. To optimize the recursive procedure, a computationally stable hybrid method is proposed which first starts the recursive computation with the transfer matrices and then, as the thickness increases, transits to the stiffness matrix recursive algorithm. Numerical results show this solution to be stable and efficient. As an application example, we calculate the surface wave velocity dispersion for a functionally graded coating on a semispace. [Copyright &y& Elsevier]

Asymptotic solutions are presented for a stationary crack normal to the boundary between two elastically mismatched solids such that the crack tip is located at the interface. The second-order term in the elastic asymptotic expansion was determined as a function of elastic mismatch for a thin cracked film on a substrate and for a thin cracked lamina between two substrates. Elastic–plastic analysis was performed using both modified boundary layer formulations and full field analyses. Analytic and numerical solutions in small strain yielding identify elastic mismatch and the T-stress as the determinants of crack tip constraint. The effect of constraint on the competition between interface failure and penetration is discussed. [Copyright &y& Elsevier]

A general law of energy release is derived for stressed heterogeneous materials, being valid from the starting moment of loading till the moment of materials fracture. This law is obtained by employing the extrapolation technique of the self-similar approximation theory. Experiments are accomplished measuring the energy release for industrial composite samples. The derived analytical law is confronted with these experimental data as well as with the known experimental data for other materials. [Copyright &y& Elsevier]

This paper examines, from the prospect of continuum analysis, the possibility for a supersonic crack growth in a solid with an upturn stress–strain relation. The stress has a linear-upturn power-law relation with the strain, resulting in an elastic modulus, and consequently a wave speed, that increase with the strain. Though appearing to be “supersonic”, the local wave speed in the crack tip vicinity of the solid with a sufficient upturn stress–strain relation exceeds the crack extension speed. A pre-request for such a supersonic crack growth is the storage of sufficient deformation energy within the solid to nurse the energy flux drawn to the crack tip that extends at an “apparent supersonic” speed. The idea is demonstrated for the simplest case, the anti-plane shear. We examine the steady-state supersonic crack growth in a hyperelastic material. The governing equation is elliptical in the crack tip vicinity but hyperbolic elsewhere. The boundary between two regions is determined with a certain extent. An asymptotic solution is constructed within the super-hardening zone. The solution connects to the hyperbolic radiation strips by weak discontinuity boundaries and to the pre-stressed frontal field by a strong discontinuity boundary. [Copyright &y& Elsevier]

A method is developed for characterising the nature of the plastic zone which develops along the boundary of any notionally complete frictionless contact but where, in practice, there is some small rounding. The approach consists of an outer asymptote, the solution for a semi-infinite square ended rigid punch, whose validity sets the upper limit to the load, and a nested inner asymptote, the solution for a semi-infinite rounded punch, which sets the lower limit to the load. The technique is applied, as an example, to a circular punch, and explicit values of the load given to ensure that the singular field characterises the local stress field to within a given degree of accuracy. [Copyright &y& Elsevier]

Hypersingular integrals of the typeIα(Tn,m,r)=−11Tn(s)(1−s2)m−1/2/(s−r)αds,|r|<1andIα(Un,m,r)=−11Un(s)(1−s2)m−1/2/(s−r)α ds, |r|<1andIα(Un,m,r)=−11Un(s)(1−s2)m−1/2/(s−r)α ds, |r|<1are investigated for general integers α (positive) and m (nonnegative), where Tn(s) and Un(s) are the Chebyshev polynomials of the first and second kinds, respectively. Exact formulas are derived for the cases α=1,2,3,4 and m=0,1,2,3; most of them corresponding to new solutions derived in this paper. Moreover, a systematic approach for evaluating these integrals when α>4 and m>3 is provided. The integrals are also evaluated as |r|>1 in order to calculate the stress intensity factors. Examples involving crack problems are given and discussed with emphasis on the linkage between mathematics and mechanics of fracture. The examples include classical linear elastic fracture mechanics, functionally graded materials, and gradient elasticity theory. An appendix, with an alternative derivation of the formulae, supplements the paper. [Copyright &y& Elsevier]

Two-scale convergence is a special weak convergence used in homogenization theory. Besides the original definitions by Nguetseng and Allaire, the paper introduces an alternative approach based on Arbogast’s idea of two-scale transform which changes a sequence of one-variable functions into a sequence of two-variable functions. Although it leads to the same convergence, it simplifies some proofs. [Copyright &y& Elsevier]

In the present work, Kaplun limits will be used to derive a generalized version of the boundary layer theory. The intermediate variable technique will be applied to the Navier–Stokes equations and, through the concept of principal equation proposed by Kaplun, a set of partial differential equations will be obtained which represents the asymptotic limit of the momentum equations as the Reynolds number approaches infinite. These equations combine the inviscid flow formulation and the classical boundary layer equations into a single and more general theory that disregards the needs for any type of viscid–inviscid interaction. The proposed formulation will be used to study the flow over a flat plate, and a quasi-similar ordinary differential equation will be deduced which represents an extension of the Blasius equation. The domain of validity of the quasi-similar equation will be analyzed and the asymptotic character of the Blasius solution, which is valid only as Reynolds number tends to infinity, will be studied. [Copyright &y& Elsevier]